Polynomials.html

Function: hall_e8[Polynomials]

Calling Sequence:

Polynomials();

Polynomials(i);

Polynomials(a);

Polynomials(i,m);

Polynomials(a,m);

Polynomials(i,m,c);

Polynomials(a,m,c);

Parameters:

i - an integer in range 1..8

a,m - dimension vectors of indecomposable modules

c - a set of dimension vectors of indecomposable modules

Description:

if the function is called with tree arguments a,m,c , of which the first and second are dimension vectors of indecomposable modules A and M respectively and the third is a set of dimension vectors of indecomposable modules , then the corresponding Hall Polynomial is returned, i.e. the number of submodules of of M isomorphic to A with factor M/A isomorphic to the direct sum C of the modules . (This number is a polynomial in the cardinality of the base field.)

Of course these Polynomials are zero unless the dimension vectors of A and C add up to the dimension vector of M. Also, A and M being indecomposable, the polynomial is known to be zero, if C should have a multiple direct summand. Thus, working with a set of dimension vectors in the third argument, which does not allow for multiple summands, is indeed sufficient.

A list of the dimension vectors of all indecomposable modules can be obtained using the function Indecomposables.

If the function is called with two parameters a,m, which are dimension vectors of indecomposable modules A and M, it returns a list which is indexed by all possible sets c of dimension vectors for which the Hall polynomial as returned by a call of Polynomials(a,m,c) is not zero. The entries of this list are the corresponding polynomials.

I the function is called with exactly one argument a, the result of the call is a list of all Hall polynomials, which is the dimension vector of an indecomposable module A, it returns a list of all nonzero Hall polynomials obtained for A with arbitrary indecomposable modules M (and arbitrary C).

Finally, if the function is called without any argument, it returns a list of all polynomials obtained as described above for indecomposable modules A and M.

In the first argument, instead of the dimension vector of an indecomposable module, an integer i in range 1..8 can be used. In this case the dimension vector of the indecomposable projective module corresponding to the vertex i of the underlying algebra, as given by a call of Projective(i); is substituted as first argument. Thus a call of Polynomials(i,...) is equivalent to a call of Polynomials(Projective(i),...) .

Examples:

> with(hall_e8):

> Polynomials();